Question

Express $2.\overline{36}+0.\overline{23}$ in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q\ne 0$.

Answer 1

Step 1: Expressing $2.3\stackrel{-}{6}$ in form $\frac{p}{q}$:

Let us assume , $x=2.3\stackrel{-}{6}=2.3666........\left(1\right)$
Multiplying both sides by$10$, we get

$10x=23.666........\left(2\right)$

Multiplying both sides of $\left(1\right)by100$, we get

$100x=236.6666........\left(3\right)$
Now , on Subtracting equation ($1$) from equation ($3$) , the repeating decimal portion cancels out.
∴ $100x-10x=213$

$\Rightarrow 90x=213$

$\Rightarrow x=\frac{213}{90}$

Step 2: To express $0.\overline{23}$ in $\frac{p}{q}$form:

Let $y=0.\overline{23}=0.232323........\left(1\right)$
Multiplying both sides of $\left(1\right)by100$, we get

$100y=23.2323........\left(2\right)$
On Subtracting equation ($1$) by equation ($2$) , the repeating decimal portion cancels out.
∴ $100y-y=23$

$\Rightarrow 99y=23$

$\Rightarrow y=\frac{23}{99}$

Step 3: Adding $xandy$

$$\begin{gathered} x+y=\frac{213}{90}+\frac{23}{99} \\ \Rightarrow x+y=\frac{23,157}{8910} \\ \Rightarrow x+y=\frac{2573}{990} \end{gathered}$$

Therefore , $2.3\stackrel{-}{6}+0.\overline{23}$ is $\frac{2573}{990}$ in $\frac{p}{q}$form.